CpSc 881: Information Retrieval

2 Using language models (LMs) for IR ❶ LM = language model ❷ We view the document as a generative model that generates the query. ❸ What we need to do: ❹ Define the precise generative model we want to use ❺ Estimate parameters (different parameters for each document’s model) ❻ Smooth to avoid zeros ❼ Apply to query and find document most likely to have generated the query ❽ Present most likely document(s) to user

3 What is a language model? We can view a finite state automaton as a deterministic language model. I wish I wish I wish I wish... Cannot generate: “wish I wish” or “I wish I”. Our basic model: each document was generated by a different automaton like this except that these automata are probabilistic.

4 A probabilistic language model This is a one-state probabilistic finite-state automaton – a unigram language model – and the state emission distribution for its one state q 1. STOP is not a word, but a special symbol indicating that the automaton stops. frog said that toad likes frog STOP P(string) = 0.01 · 0.03 · 0.04 · 0.01 · 0.02 · 0.01 X(0.8X0.8X0.8X0.8X0.8X0.8X0.2) =

5 A different language model for each document frog said that toad likes that dog P(string|M d1 ) = 0.01 · 0.03 · 0.04 · 0.01 · 0.02 · 0.04 · = 4.8· P(string|M d2 ) = · 0.03 · 0.04 · · 0.04 · 0.04 · 0.01 = 3.84 · P(string|M d1 ) > P(string|M d2 ) Thus, document d 2 is “more relevant” to the string “frog said that toad likes that dog” than d 1 is.

6 Using language models in IR  Each document is treated as (the basis for) a language model.  Given a query q  Rank documents based on P(d|q)  P(q) is the same for all documents, so ignore  P(d) is the prior – often treated as the same for all d  But we can give a prior to “high-quality” documents, e.g., those with high PageRank.  P(q|d) is the probability of q given d.  So to rank documents according to relevance to q, ranking according to P(q|d) and P(d|q) is equivalent.

7 Where we are  In the LM approach to IR, we attempt to model the query generation process.  Then we rank documents by the probability that a query would be observed as a random sample from the respective document model.  That is, we rank according to P(q|d).  Next: how do we compute P(q|d)?

8 8 How to compute P(q|d)  We will make the same conditional independence assumption as for Naive Bayes. (|q|: length ofr q; t k : the token occurring at position k in q)  This is equivalent to:  tf t,q : term frequency (# occurrences) of t in q  Multinomial model (omitting constant factor)

9 Parameter estimation  Missing piece: Where do the parameters P(t|M d ). come from?  Start with maximum likelihood estimates (as we did for Naive Bayes) (|d|: length of d; tf t,d : # occurrences of t in d)  As in Naive Bayes, we have a problem with zeros.  A single t with P(t|M d ) = 0 will make zero.  We would give a single term “veto power”.  For example, for query [Michael Jackson top hits] a document about “top songs” (but not using the word “hits”) would have P(t|M d ) = 0. – That’s bad.  We need to smooth the estimates to avoid zeros.

10 Smoothing  Key intuition: A nonoccurring term is possible (even though it didn’t occur),... ... but no more likely than would be expected by chance in the collection.  Notation: M c : the collection model; cf t : the number of occurrences of t in the collection; : the total number of tokens in the collection.  We will use to “smooth” P(t|d) away from zero.

11 Mixture model  P(t|d) = λP(t|M d ) + (1 - λ)P(t|M c )  Mixes the probability from the document with the general collection frequency of the word.  High value of λ: “conjunctive-like” search – tends to retrieve documents containing all query words.  Low value of λ: more disjunctive, suitable for long queries  Correctly setting λ is very important for good performance.

12 Mixture model: Summary  What we model: The user has a document in mind and generates the query from this document.  The equation represents the probability that the document that the user had in mind was in fact this one.

13 Example  Collection: d 1 and d 2  d 1 : Jackson was one of the most talented entertainers of all time  d 2 : Michael Jackson anointed himself King of Pop  Query q: Michael Jackson  Use mixture model with λ = 1/2  P(q|d 1 ) = [(0/11 + 1/18)/2] · [(1/11 + 2/18)/2] ≈  P(q|d 2 ) = [(1/7 + 1/18)/2] · [(1/7 + 2/18)/2] ≈  Ranking: d 2 > d 1

14 Exercise: Compute ranking  Collection: d 1 and d 2  d 1 : Xerox reports a profit but revenue is down  d 2 : Lucene narrows quarter loss but decreases further  Query q: revenue down  Use mixture model with λ = 1/2  P(q|d 1 ) = [(1/8 + 2/16)/2] · [(1/8 + 1/16)/2] = 1/8 · 3/32 = 3/256  P(q|d 2 ) = [(1/8 + 2/16)/2] · [(0/8 + 1/16)/2] = 1/8 · 1/32 = 1/256  Ranking: d 2 > d 1

15 Vector space (tf-idf) vs. LM The language modeling approach always does better in these experiments but note that where the approach shows significant gains is at higher levels of recall.

16 LMs vs. vector space model (1)  LMs have some things in common with vector space models.  Term frequency is directed in the model.  But it is not scaled in LMs.  Probabilities are inherently “length-normalized”.  Cosine normalization does something similar for vector space.  Mixing document and collection frequencies has an effect similar to idf.  Terms rare in the general collection, but common in some documents will have a greater influence on the ranking.

17 LMs vs. vector space model (2)  LMs vs. vector space model: commonalities  Term frequency is directly in the model.  Probabilities are inherently “length-normalized”.  Mixing document and collection frequencies has an effect similar to idf.  LMs vs. vector space model: differences  LMs: based on probability theory  Vector space: based on similarity, a geometric/ linear algebra notion  Collection frequency vs. document frequency  Details of term frequency, length normalization etc.

18 Language models for IR: Assumptions  Simplifying assumption: Queries and documents are objects of same type. Not true!  There are other LMs for IR that do not make this assumption.  The vector space model makes the same assumption.  Simplifying assumption: Terms are conditionally independent.  Again, vector space model (and Naive Bayes) makes the same assumption.  Cleaner statement of assumptions than vector space  Thus, better theoretical foundation than vector space  … but “pure” LMs perform much worse than “tuned” LMs.